The field of non-imaging optics has long sought a method to design optical surfaces that transform an incident light emitted by a light source into an arbitrary illumination pattern. In the last two decades, substantial progress has been made for the zero-étendue case, an idealization where light rays are exactly parallel or exactly diverging from a single point. This idealization allows a one-to-one correspondence between rays in the emitted light and rays in the target illumination pattern. This one-to-one correspondence reduces the design problem to determining an optical surface whose reflections or refractions implement a one-to-one mapping between the spatial density of rays in a cross-section of the emitted light and the spatial density of rays in the target illumination pattern. If a smooth mapping is possible between the initial and target densities, which is almost always the case for the zero-étendue systems, then that mapping can be found using the methods borrowed from the field of optimal mass transport. The resulting optics can produce very complicated illumination patterns, for example, projecting photographic images. These optical surfaces are generally denoted as freeform optical surfaces, simply because their shapes are more complicated that any of the simple algebraic surfaces typically associated with lenses and mirrors.
In reality, the zero-étendue light source does not exist. Practical light sources, e.g., light-emitting diodes (LED), have spatial extent, i.e., light rays are emitted from an area, not a point, and these rays cross during their propagation, making one-to-one mappings impossible, and pushing the problem outside the scope of what optimal mass transport can solve. If a freeform optical surface is illuminated by a spatially extended light source, the resulting illumination pattern is significantly blurred, much as a shadow on a cloudy day becomes soft and indistinct. According to the second law of thermodynamics, this blurring is inescapable, so freeform optics for spatially extended light sources are typically designed to achieve approximately uniform illumination in some bounded area surrounded by soft blurry illumination fall-offs.
For example, a method of simultaneous multiple surfaces (SMS) achieve uniform illumination by directing rays of light from the edge of a spatially extended light source to desired target points. In such a way, the rays of light form some unknown but acceptable density; however, the edges of the resulting illumination pattern are still blurred.
At the best of our knowledge, currently there are no methods that can transform incident light from the spatially extended light source into a target illumination pattern with sharp edges. However, such a method can be beneficial for a number of optical applications, such as optics for signage illumination and optics for headlight of a vehicle.